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The Geometry of Supersymmetric Non-Linear Sigma Models in D ≤ 2 Dimensions

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The Geometry of Supersymmetric Non-Linear Sigma Models in D ≤ 2 Dimensions The geometry of supersymmetric non-linear sigma models in D ≤ 2 dimensions Malin G¨oteman February 19, 2008 Master thesis Supervisor: Ulf Lindstr¨om Department of Physics and Astronomy Division of Theoretical Physics University of Uppsala 1 Abstract After a review of the two-dimensional supersymmetric non-linear sigma models and the geometric constraints they put on the target space, I focus on sigma models in one dimension. The mathematical framework in terms of supersymmetry and complex geometry will also be studied and reviewed. The geometric constraints arising in D = 1 are more general than in D = 2, and can only after some assumptions be reduced to the well known geometries arising in the two dimensional case. 2 Contents 1 Introduction 4 2 Sigma models 5 2.1 The bosonic sigma model in D =2....................... 5 2.2 The bosonic sigma model in D =1....................... 6 3 Supersymmetry and superfields 8 3.1 The supersymmetry algebra . 8 3.2 Superspace and superfields . 10 4 Supersymmetric sigma models 12 4.1 Supersymmetric sigma models in D =2 .................... 12 4.2 Supersymmetric sigma models in D =1 .................... 12 5 Complex geometry 14 5.1 Complex structures . 14 5.2 Generalized complex structures . 15 6 Geometry of supersymmetric sigma models in D = 2 17 6.1 Complex geometry realized in D = 2 sigma models . 17 6.2 Generalized complex geometry realized in D = 2 sigma models . 18 7 Geometry of supersymmetric sigma models in D = 1 20 7.1 Geometry of D = 1 supersymmetric sigma models . 21 7.2 Constructing N = 1 model in D = 1 by dimensional reduction . 23 7.2.1 Reduction directly from a N = (1, 1) model . 23 7.2.2 Reduction via manifest N = 2a model in one dimension . 25 7.2.3 Reduction from N = (2, 0) model . 27 7.3 Geometry of D = 1 models compared to higher dimensions . 28 8 Summary and conclusions 30 A Appendix 32 A.1 Field equations for N = (1, 1) susy sigma model in D = 2 . 32 A.2 Manifest invariance of N =(1, 1) sigma model under N =(1, 1) transformations 32 A.3 Integrability conditions for distributions . 33 A.4 Dimensional reduction of manifest N = (2, 2) sigma model in D = 2 . 34 3 1 Introduction Non-linear sigma models provide a link between supersymmetry and complex geometry. The number of supersymmetries imposed on the sigma model determine the geometry of the target space, as was first realized in [3] and developed in [4], [5], [7], among others. In dimension D = 2, which is the dimension central for string theory, one supersymmetry implies no restriction on the target manifold, whereas two supersymmetires require a K¨ahler manifold and four supersymmetries require hyper-K¨ahler geometry. This has been described in detail in [7], [20], [31], [32] and will be studied and reviewed in section 6. The mathematical framework in terms of generalized complex structures was developed in [24] and [25]. Under the two assumptions, that the kinetic part of the Lagrangian depends only on µ a ν µ the metric as ∼ gµν(φ)∂aφ ∂ φ , and that the fields φ are functions of time and at least one spatial coordinate, three classes of supersymmetric sigma models are known: generic, K¨ahler and hyper-K¨ahler [29]. The first assumption can be extended by introducing an anti-symmetric B-field in addition to the metric, which was realized in [7] and will be reviewed in section 2.1 and 6. Obviously, the second assumption is automaticly relaxed when studying one-dimensional sigma models, since the fields don’t depend on spatial coordinates per definition. Therefore, in D = 1, even a larger variety of supersymmetric sigma models can be constructed, as we will see in section 4.2. The bosonic sigma model is derived in section 2 and its supersymmetric extension in section 4. Supersymmetry, (generalized) complex geometry and further mathematical framework needed for the study of non-linear sigma models are reviewed in section 3 and 5. In section 7, I focus on the geometry of target space arising from supersymmetric non- linear sigma models in dimension D = 1. After a review of what is known in the area, I discuss some of these results in more detail in section 7.1 and in section 7.2 I explicitly construct a one-dimensional sigma model by dimensional reduction from two-dimensional sigma models. In section 7.3, the geometry arising on the target space by supersymmetric sigma models in one dimension is compared with higher dimensional cases. The manifolds of the one dimensional sigma models have a more complicated structure, and can only after certain assumptions be reduced to the well-known geometries that appear in dimension D = 2 [12]. Also, in D = 1, there is some flexibility when deriving the constraints imposed on the target space [22]. In one dimension, there is less space-time symmetry, which implies that one can construct more general Lagrangians than in higher dimensions. Some supersymmetric D = 1 sigma models feature target space geometries which cannot be reproduced by direct dimensional reduction from higher dimensional models, a fact which make them an interesting subject to study. The one dimensional sigma models have many applications, such as describing the geodesic motion in the moduli space of black holes [17] and being the model for supersym- metric quantum mechanics, which arises in the light cone quantization of supersymmetric field theories. For clarity, most of the longer calculations have been omitted or relegated to the appendix. 4 2 Sigma models A sigma model is a set of maps Xµ : P → T , where ξi ∈ P, i = 1, ..., D are the coordinates on the D-dimensional parameter space P and Xµ, µ = 0, ..., d − 1 are the coordinates in the d-dimensional target space T , and an action giving the dynamics of the model. 2.1 The bosonic sigma model in D = 2 Although in no way fundamental, it is interesting that the action describing the 2-dimensional bosonic sigma model can be derived from a classical string. The potential energy of the string depends on its tension T , and setting c = 1, the mass density is equal to the tension and we get an action of the form Z S = −T dA (2.1) Denote the Minkowski metric of the target space by ηµν = diag(−1, 1, ..., 1) and let γab be the induced metric on the world surface, 2 µ ∂X µ ν ∂X ∂X µ ∂X ∂X 1 2 1 γ = η = dξ dξ dξ . (2.2) ab a b µν µ 2 ∂ξ ∂ξ ∂X ∂Xµ ∂X dξ1 dξ2 dξ2 We require the action of the theory to be invariant under diffeomorphisms. Under a coordinate transformation, the invariant volume element is in general given by the so p √ called proper volume dV = d ξ − det γab. To see this, we note that under a coordinate h ∂ξ0m i transformation, writing γ := det γab and the Jacobian matrix Λ := ∂ξµ , √ √ q p −γ dpξ 7→ det Λdpξ and −γ 7→ − det((Λ−1)T γ Λ−1) = −γ(det Λ−1)2 = . ab det Λ (2.3) 2 √ Hence, the area element on the world sheet is given by dA = d ξ − det γab and we get the Nambu-Goto action [26] Z Z q 2 p 2 µ 2 2 2 S = −T d ξ − det γab = −T d ξ (∂aX ∂bXµ) − (∂aX) (∂bX) , (2.4) where a, b ∈ {1, 2} are the indices for the parameters ξa on the world surface. The difficulties of quantizing this action motivates the introduction of the classically equivalent Polyakov action [1], [2] T Z √ S = − d2ξ −hhabγ , (2.5) 2 ab where h := det hab, hab being defined as the independent metric of the world sheet. The fact that the Polyakov action is equivalent with the Nambu-Goto action can be seen by varying the action (2.5) with respect to hab: T Z √ 1 δS = − d2ξ −hγ − h hcdγ δhab. (2.6) 2 ab 2 ab cd 5 √ cd √ cd Requiring this to be zero gives 2γab = habh γcd which in turn implies 2 −γ = h γcd −h. Inserting this into the Polyakov action (2.5) recovers the Nambu-Goto action (2.4). By a theorem by Hilbert, for any 2-dimensional surface with metric hab we can choose conformal coordinates√ in which the metric takes the diagonal form h12 = h21 = 0, h11 = −h22 so that − det hab = h11. In this gauge the Polyakov action (2.5) takes the simplified form T Z √ S = − d2ξ −hhabγ 2 ab T Z = − d2ξh (h11γ + 0 + 0 − h11γ 2 11 11 22) T Z ∂Xµ ∂Xν ∂Xµ ∂Xν = − d2ξ η − η 2 µν ∂ξ1 ∂ξ1 µν ∂ξ2 ∂ξ2 T Z = d2ξη ∂ Xµ∂aXν. (2.7) 2 µν a For a target space with curvature, the Minkowski metric ηµν is replaced by a general metric Gµν, and finally we arrive at the bosonic non-linear sigma model action T Z S = dτdσ∂ Xµ∂aXνG (X). (2.8) 2 a µν In D = 2, we can include an anti-symmetric tensor Bµν in the background. Using light-cone coordinates, x±± = √1 (ξ1 ± ξ2) the action thus takes the simple form 2 1 Z h i Z S = d2x ∂ Xµηab∂ XνG (X)+ab∂ Xµ∂ XνB = d2x∂ XµE ∂ Xν. (2.9) 2 a b µν a b µν ++ µν = where Eµν = Gµν + Bµν and we for simplicity skipped the factor T . The field equations are obtained from δS = 0: µ (0)µ µ ν τ ∂++∂=X + (Γντ + Tντ )∂++X ∂=X = 0, (2.10) or in shorter notation, (+) µ ∇++ ∂=X = 0.
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